Real Analysis Theorems

E is bouded if…

E is both bounded above and below. This means there exist real numbers, say $m$ and $M$, such that for all elements x in the set E, we have m <= x <= M.

The Axiom of Completeness

Every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers.

R is ordered

There exists an ordered field R that satisfies the Axiom of Completeness, and Q is a subset of R.

Suppose A is a subset of R and is nonempty+bdd below, s.t. A = {-a : a in A}. Then…

the infimum of A is equal to the negative supremum of -A. Thus inf(A) exists.

The Archimedean Principle

For all x,y in R where x > 0, there exists some natural number n such that x*n > y.

The Density of Q in R

For all a,b in R, where a < b, there exists a rational number between a and b.

Injectivity

A function f: A→B is injective, or one-to-one, if x1 /= x2 implies f(x1) /= f(x2).

Surjectivity

A function f:A→B is surjective, or onto, if f(A) = B, or for every element of the range, there is some element in the domain such that f(domain element) = range element.

Bijectivity

If a function f:A→B is both injective and surjective, it is bijective i.e. for all y in the range, there is ONE x in the domain s.t. f(x) = y.

Equivalent / Same Cardinality

If there exists a bijection f:A→B, we say A ~ B, thus A and B have the same cardinality.

Transitivity of Equivalences

If A ~ B and B ~ C, then A ~ C, i.e. if f:A→B bijective and g:B→C bijective, then g(f(A)):A→C bijective.

Finite set

A set is finite if it is empty or if a bijection exists between the set and a finite sequence of natural numbers, up to n, i.e. A ~ {1,2,3,4,…,n}.

Countability

A set S is countable if S ~ N, i.e. if a bijection exists between S and N, f:S→N bijective.

Uncountability

A set B is uncountable if it is infinite and not countable.

Countable union of countable sets

The countable union of countable sets is also countable.

Algebraic numbers

A real number is said to be algebraic if there exists a natural number n and a0, a1, a2, …, a_n (all nonzero integers) such that a_n*x^n + a_{n-1}*x^{n-1} + a_{n-2}*x^{n-2} +…+a_1*x + a_0 = 0.

Subset of countable sets

If A and B are sets, and B is a subset of A, then A being countable implies B is at most countable (finite or countable).

Cantor’s Theorem

There does not exist a surjectio from a set A to its power set P(A).

Nested Interval Property

Let {I_n} be a set of closed, nested intervals, i.e. I_n = [a_n, b_n] and I_1 is in I_2 is in I_3 … . Then there exists some real number epsilon such that epsilon is in I_n for all natural numbers n. In other words, the intersection of all I_n’s is nonempty.

Sequences

A sequence of real numbers is a function f:N→R.

Convergence

A sequence of real numbers converges to some real number x if for all epsilon positive, there is some natural number N such that |xn - x| < epsilon for all n >= N.

Epsilon neighborhood of a

Take some real number a, and fix epsilon > 0. The epsilon-neighborhood of a is denoted by V_epsilon(a) = {real numbers x | |x-a| < epsilon}

A sequence (xn) converging implies…

(xn) is bounded.

Convergence with epsilon neighborhood

(xn)→x iff every epislon neighborhood of x contains xn for all but a finite number of natural numbers N.

Uniqueness of Limits

If (xn)→x and (xn)→x’, then x = x’.

The real numbers are uncountable.

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